SLL 80.686 copy

NRL Memorandum Report 3587

Stimulated Backscattering From Relativistic Unmagnetized Electron Beams

P.SPRANGLE

Plasma Physics Division

and

A. T. DROBOT CO

Science Applications, Incorporated McLean, Virginia 22101

CXI

February 1978

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Accession Number: 3909

Publication Date: Feb 01,1978

Title: Stimulated Backscattering from Relativistic Unmagnetized Electron Beams

Personal Author: Sprangle, P.; Drobot, A.T.

Corporate Author Or Publisher: Naval Research Laboratory, Washington, DC Report Number: NRL MR 3587 Report Number Assigned by Contract Monitor: SLL 80 686

Comments on Document: Archive, RRI, DEW

Descriptors, Keywords: Stimulate Backscatter Relativistic Unmagnetized Electron Beam Nonlinear Saturation Wave-Wave Function Interaction Incident Pump Frequency Level Efficiency

Pages: 00026

Cataloged Date: Nov 27, 1992

Document Type: HC

Number of Copies In Library: 000001

Record ID: 25255

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REPORT DOCUMENTATION PAGE 1. REPORT NUMBER

NRL Memorandum Report 3587

2. GOVT ACCESSION NO

4. TITLE fand Subtitle)

STIMULATED BACKSCATTERING FROM RELATIVISTIC UNMAGNETIZED ELECTRON BEAMS

READ INSTRUCTIONS BEFORE COMPLETING FORM

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7. AUTHOR^; 8. CONTRACT OR GRANT NUMBERfaJ

P. Sprangle and A. T. Drobot^

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Naval Research Laboratory Washington, DC 20375

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Science Applications, Inc., McLean, VA 22101

19. KEY WORDS (Continue on reverse aide It neceaaary and Identify by block number)

Scattering Relativistic Electron Beams Non-Linear Saturation Wave-Wave Interaction

20. ABSTRACT (Continue on reverse aide It neceaaary and Identify by block number;

Analysis of stimulated scattering of a high frequency incident pump wave from an unmagne- tized relativistic electron beam is presented. The backscattered radiation frequency can be enhanced by the factor 4-y2 over the incident pump frequency where 7Qis the relativistic factor of the electron beam. The linear growth rates associated with the wave-wave and wave-particle modes of scattering are examined for a number of different pump amplitude regimes. Estimates for the scattering efficiency are presented for the wave-wave scattering process.

DD 1 JAN"73 1473 EDITION OF 1 NOV 65 IS OBSOLETE S'N 0102-LF-014-6601 Unclassified

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CONTENTS

Section I. Introduction 1

Section II. Dispersion Relation 4

Section III. Saturation Levels and Efficiencies 18

References 23

in

STIMULATED BACKSCATTERING FROM

RELATIVISTIC UNMAGNETIZED ELECTRON BEAMS

Section I. Introduction

Stimulated emission of backscattered radiation from intense relativistic electron beams

has received considerable interest in the past few years. The primary reason for this interest

lies in the fact that radiation backscattered from relativistic electron beams can undergo a

dramatic frequency increase and is readily tunable over a wide frequency range. Hence, these

scattering mechanisms, which rely on relativistic electron beams, may soon lead to a new class

of submillimeter and infrared generating devices which could find application in such areas as

radar, plasma heating, diagnostics, isotope separation and laser pellet fusion.

Analyses of the scattering phenomena have been carried out using both, a quantum

mechanical formalism1 ~4 as well as a classical approach.5-9 In these theories, the incident

pump field has taken various forms such as periodic static fields and traveling electromagnetic

waves. Numerical simulations of the scattering processes have shown that the efficiency of

converting electron kinetic energy into electromagnetic energy can be as high as 30% under

certain conditions.10,11 The frequency enhancement can be viewed as a double doppler upshift

of the incident pump wave. An incident electromagnetic pump field at frequency w0, propagat-

ing antiparallel to a relativistic electron beam with speed v0 will backscatter into a frequency

~ (1 + v0/c)27oW0 where y0 = (1 - (v0/c)2) ~1/2. In the case of a periodic static pump

Manuscript submitted August 11, 1977.

SPRANGLE AND DROBOT

field of period /, the frequency of the backscattered wave will be approximately given by

(1 + v0/c)y%v0(2ir/l). The upshifted frequency can be easily varied by changing the energy

of the electron beam. Experiments at Stanford University have demonstrated laser action at

wavelengths of 10.6 ^m as well as 3.4 /u.m using a static periodic pump. The gain in these ex-

periments was relatively low: 7% increase in power was achieved in a 5.2 m interaction length

at 10.6 /Lim and in an oscillator experiment only 0.01% of the electron beam energy was con-

verted into radation. Recent experiments at the Naval Research Laboratory using a traveling

electromagnetic pump field have produced power levels of 1.5 MWs at 0.5 mm with an overall

efficiency of 0.01%. At Columbia University experiments 12,13 employing a static periodic mag-

netic pump have resulted in megawatts of scattered radiation at wavelengths in the neighbor-

hood of 1 mm. Scattering experiments using relativistic electron beams are also in progress at

the Ecole Polytechnique in France.14

The two principal types of scattering processes in which an incident pump field is back-

scattered off an electron distribution into a transverse wave are wave-wave (Raman) and

wave-particle (Compton) scattering.15 ~19 In general, these two scattering modes are present

simultaneously; however, the wave-wave process dominates if the incident pump wavelength

in the electron beam frame is much greater than the Debye wavelength. Scattering then takes

place off collective plasma oscillations. On the other hand, wave-particle scattering dominates

when the pump wavelength is comparable to or smaller than the Debye wavelength. In this

situation, scattering takes place off shielded or "dressed" particles . This paper will address both,

wave-wave and wave-particle scattering.

The physical mechanism responsible for the instability of the backscattered electromag-

netic wave, i.e., stimulated emission, can readily be described classically in the beam frame. In

what follows, quantities in the beam frame will be written with primes. In the beam frame we

stipulate that the existing electron equilibrium is perturbed by a low frequency density wave in

NRL MEMORANDUM REPORT 3587

the absence of an external magnetic field. Only waves propagating along the z axis, i.e., direc-

tion of beam velocity in the laboratory frame, will be considered. The electrostatic perturbation

at frequency and wavenumber (CD\\ , k\\ ) need not be an eigenmode of the electron distribu-

tion. The introduction of a large amplitude high frequency incident pump, £0, at (a»0, k0)

forces the electrons to oscillate at a frequency w0 in the direction along E0 with a maximum

velocity given by vov = | e\ EJ (moa>0). This transverse oscillation velocity, vos perpendicular

to k0, couples to the density wave, thus inducing transverse currents at frequency

w+ = w|| ± w0 and wave numbers k± = k\\ ± k0. These currents now generate new

electromagnetic waves at (to±, A± ). The generated or scattered electromagnetic field consists

of backscattered waves propagating antiparallel to the incident pump wave. Forward scattered

waves are also induced, but will not be considered because they are down shifted in frequency

and also have a much smaller growth rate than the backscattered radiation. The pump and

backscattered wave couple through the v' x B' term in the Lorentz force equation resulting in

a longitudinal force at (o»n , An ). This induced longitudinal force, also called the ponderomo-

tive or radiation pressure force, if properly phased will reinforce the originial density wave. The

backscattered electromagnetic wave is, therefore, unstable resulting in stimulated emission of

radiation. It should be noted that in the beam frame, the pump frequency is usually much

greater than the frequency of the longitudinal wave, |w0| » |WJ||.

For a cold electron beam the electrostatic wave is an eigenmode of the system, |wn | is

roughly equal to the electron plasma frequency, w = (4TT-| e\ 2n0m0) l'2 and the scattering

process is referred to as Raman scattering. However, if the pump strengh is sufficiently strong,

the frequency of the electrostatic wave is modified by the pump field and is greater than the

plasma frequency. In this regime the scattering process is called modified Raman scattering. In

either case, the phase velocity of the electrostatic wave is far removed from the electron

velocity, |to||/A|| | >> vth, where vth is the electron thermal velocity; therefore, they are

SPRANGLE AND DROBOT

referred to as nonresonant, wave-wave or collective scattering modes. If the electron beam is

sufficiently thermal so that the phase velocity of the electrostatic wave is comparable to the

electron velocity, a resonance between the wave and particles results. This regime is called

Compton scattering, resonant wave-particle scattering or inverse nonlinear Landau damping.

Here the nonlinear coupling between the pump wave and scattered electromagnetic wave in-

duces a longitudinal wave with a phase velocity comparable to the electron thermal velocity,

\<*>\\lk\\ | - vlh.

Section II. Dispersion Relation

In this section equations describing the coupling of the incident pump wave and the scat-

tered electromagnetic and scattered electrostatic waves are derived. The large amplitude in-

cident pump field is assumed to be linearly polarized in the x direction with frequency QJ0, and

wavenumber k0 = k0e.. Only spatial variations along the z axis will be considered. The pump

field is incident upon a system of electrons which are electrostatically as well as magnetically

neutral. The model is depicted in Fig. (1) and the analysis is fully relativistic and is performed

in the laboratory frame of reference. The electromagnetic field of the incident pump wave is

chosen to be of the form

E0 (z, t) = E0 cos (k0z — (»0t)ex,

ck0 B0 (z, /) = E0 cos (k ,z - (o0t)e ,

(la)

where E0 is the electric field amplitude and the direction of the axial Poynting flux along the z

axis is given by the sign of <o0/k0. The form of the scattered electrostatic wave is

En = E\\ cos (k\\ z - co11 t + 0|| )e., (2)

where 4>\\ is the phase of the longitudinal field with respect to the pump field. The scattered

electromagnetic field is chosen to be of the form

4

NRL MEMORANDUM REPORT 3587

Es = £ E± cos (k±z — w±t + <j)±)ex, +, -

B,

where k± = k\\ ± k0, w.

the sum is taken over +, ■

ck. E+ cos (k+z — o) + t + (f>+ )ev,

(3a)

WH ± (D0, 0± is the phase with respect to the pump wave and

Figure 1. Schematic of Backscatter- ing Off a Relativistic Electron Beam.

Kez. a>o

INCIDENT E.M. PUMP

A

A

/^

k_ ~ Mk„ez, Cü_ = —Mcu0

BACKSCATTERED E.M. WAVE

kM = Mkoe2, Q)M = -Mcu0

BACKSCATTERED LONGITUDINAL WAVE

RELATIVISTIC ELECTRON BEAM

MEd+VctV

The evolution of the electron distribution is described by the relativistic Vlasov equation

9 , 9 , f(z, v, t) = 0, (4)

where vn =v • e. is the component of velocity along the z axis, L = |e|/w0(E +

v x B/c) • d/du, E = E0 + En + Ev, B = B0 + Bs, u = y v is the normalized momentum,

y = (1 — ß2) ~''2 and ß = v/c. In order to obtain the currents which drive the scattered

fields we use a perturbation expansion to find the distribution function /(z, v, t) in terms of

the scattered fields. Since the operator L consists of the perturbing fields, which include the

pump, we may expand / in powers of the perturbating field amplitudes, that is

= /(0) , /•(!) /' =fW' +fU' +fu> + (2)

(5)

SPRANGLE AND DROBOT

where

BJ (0)

Bt

B , B

o,

(/;) = ffUi-l) fw> = Lf

(6)

(7)

and n = 1, 2, .... In what follows the equilibrium distribution function described in (6) is

chosen to have the form

f(0)(u) =/io8G/v)8 0/v)so(i/|| ), (8)

where n0 is the ambient electron density, 8 (w,) is a delta function and f g0(u\\ )du\\ = 1.

That is, the equilibrium distribution function is chosen to be cold in momentum space

transverse to the direction of wave propagation while having a velocity spread parallel to the

direction of wave propagation. It proves convenient to write the operator, L as the sum of two

terms, one involving the pump field and the other scattered fields, that is, L = L0 + Ls where

= i£L m.

L«=n lE»cos (V -<V)0J.

£|l cos (A-|| z - ton / + 0|| ) + £ E± cos (k±z -w±t + 0± )#H

and

*o =

<K =

*o B Vxko

°>o Bux ' wn du\\

</'± 9 9wA.

, v.v^± 9 w± co± 9w||

"/'o = wo _vll^0- </■'+= w + — V|| A-_

•/'II = wll - V|| A-||

(9a-g)

NRL MEMORANDUM REPORT 3587

The perturbing density and currents which drive the backscattered waves are given by

"e = I n/fl)(z,/), n =1

and,

J = £ J("}(z, r),

where «e(/,) = J*/^ (z, u, t)d3u and J (/,) = - \e\ f (u/y )fM (z, u, t)d3u. The response

current J drives the fields in Eqs. (1), (2) and (3) through the wave equation:

V2E - c _292E/3/2 =4nc ~2dJ/dt + V (V • E).

Solving Eq. (7) for/^'Cz, u, /), the first order particle and current density take the form

II «0/C||

" ° ) ^11 • *ll ) = _ y~ X Hi , *n )£|| sin (/C|| z - a>|| r + <£|| ),

/||(1) (con , A.',, ) =-T^r x^ll ' *H ^H sin ^11 z ~~ WH ' + *M *'

y"'t-t')--4.<"i„> ^"- <V —.»■

n. o)2 £± ./+ (co + , £ + ) = - -— sin (k+z - o+t + <j>+ ),

477<r|| > 01 ± " - " (10ad)

where mp = (4ir\e\ 2n0/m0) 1/2, \ (c>\\ • k\\ ^ = ^j/k\\ ) / du{l (Bg0(uu )/du{l )/4>ll is the

electron susceptibility, <yM > -1 = J duu g0 (wM )/yn and yM =(1 +W|2/c2)1/2. For a

cold electron beam, g0(w|| ) = 8(//|| — w0), the electron susceptibility is then given by

Xcoid = ~ ^olyl y (wll — V0AT|| ) 2. The arguments of the quantities on the left hand side

of Eqs. (10) denote the frequency and wavenumber of the quantities. Using (lOb-d) in the

wave equation the linear dispersion relations for the pump, scattered electrostatic and elec-

tromagnetic waves are respectively:

SPRANGLE AND DROBOT

2 _ „2a < -ClK -W,7/<T|| > =0,

1 +x(u>||,*|, ) =0,

2 - Aa -,.,2 toi ~ CLk vjKyw > =0.

Evaluating the second order particle and current density gives:

E. *ll En «(2)(a»„,*,i ) = -^- ~° cos (A|| z - öj|| / + <£ + )

cos (An z — ton / + <f) _ ) x(toH,/c|, ),

■'ll(2)^l|.*||) ton A: 11*11 \e\E„

cos (A'n z — co 11 / + t/> + )

COS (All Z — ton / + c/> _) X(to|| , An )

■(2) Jr(*>o.K) =T- I Hf,

877 H—L w()w.

42)(to + ,A+)

£|l cos (k0z -to0/± (0± -0M ))x(toN , AM ),

*ll \e\En

877 w0co„

(lla-c)

(12a-d)

£|l cos (A±z - w±/ + </>n )x(w|| , A,, ),

where x(cou , A,, ) = (co^/A,, ) / </tf|| (dg0(ul{ )/3W|| )/(y„ ,,,„ ). Note the difference in

the definitions of x and *. It is necessary to find the third order transverse current density at

(to±, k± ) in order to recover the wave-particle scattering. The third order current density at

(co + , A . ) is

NRL MEMORANDUM REPORT 3587

j£H<o±,k±) = \e\\ \\e\E0] 2 H<»o E + 4m0 >V0

2 A± <"VW± w +

E_ sin (k±z — <o±t + # _)

O) — (13)

where

a>; *± =

+ <h

Sduw So (»II)

rii*i^±

2-».,. 2

(/C||W0 +to\\k0) "II 6>±

A:± -1' 7|| C2

(Ar,2| -a>,2Jc2)^±^0 + (k0k± - w0w±/c2)>p

(k0kn + w0(oN/c2)(a>± +M||*±/y||)

(14)

Now substituting the currents Jl{ (a»,, , kn ) = 7||(1) + J\\2),J± (w±, k± ) = 7|n + y_ji2) +

7|3) and 70(w0, /t0) = 70(1) +/0

(2) into the wave equation for EM , E* and E0 we obtain

(1 + x(w|| , fc|| ))£|| e e $|l ;(k|| z -&i|| i) \e\En

k\\ X(w|| ,*|| )

(l) 4_ Ct) _

/(A:|| Z — <U|| /)

D±(co±, k±)E±eJ±e \e\En

X(0)|| , AT,i ) ± X.

(1 + x(w|| , A:,| )) w

x(w|| , A:,, )

„ '*ll i(k±z -w±l) En e e ± ,

D.iu.JcjEie1**''-"» = -f-lf *„^,|X(»||.*„) 2 m0

£+ /(«+ -*|| ) E _ -/(<£_ -*|| ) /(^„Z -O)0t)

(15a-c)

Combining Eqs. (15), the following dispersion relation for the scattered waves is obtained

SPRANGLE AND DROBOT

(1 + x) = \e\E0

2m0a>0 kU

(X ~ (X+/x)(l + X )*»<,/<» + ) (X + (X_/x)(l +x)a>0/ft»-) Z>. D_ • (16)

The dispersion relation in Eq. (16) describes the relationship between wN and JtM in the

laboratory frame. It is convenient, however, to transform Eq. (16) to the beam frame where the

average electron momentum is zero; < «,, > =0. Beam frame quantities will be denoted by

primes. In order to simplify the dispersion relation in the beam frame we assume that the fre-

quency of the electrostatic wave is much smaller than either the pump or scattered electromag-

netic wave, |w|| | « |w0|, and hence u'0 = ± w±. It is easy to see that this is an excellent

approximation in the beam frame. The expression for \ ± and x in the prime frame can be ap-

proximated by

*; -x'

x'-x' (17a,b)

By using Eqs. (17a, b) and assuming the electron thermal velocity is nonrelativistic in the

beam frame, Eq. (16) reduces to the rather simple form

(1 +x') = - (vM/2)2C*n)22x' 1 + l

D. D. (18)

where

:'= =*-/*, dg0 (vN )/9v|

£>; = (cO2 -c2(k'):

vll*ll

JP •

■ _ u\<

«}p = (4ir|e| 2n0/m0)1/2.

Note that m0 is the electron rest mass and hence is the same in all frames. Equation (18) is

the dispersion relation for waves scattered parallel or antiparallel to the incident pump wave off

a cold (Raman Scattering) or thermal (Compton Scattering) distribution of particles in the

10

NRL MEMORANDUM REPORT 3587

beam frame. Before examining the different scattering modes given by Eq. (18), the electron

susceptibility is written in terms of the standard plasma dispersion function, Z(f') =

77- ~1/2 J* dxexpi -x2)/(x -£') for Imf > 0. In terms of Z(£'), the suceptibility is

X = kD

*ll a +rz(r» = -j kD

h

2

(19)

where kD = oip/vrt is the Debye wavenumber, £' = (o>n //cN )/(V2 v,/;), and v,/( is the ther-

mal electron velocity defined by g0 (vM ) = (J2n v'lh) _1 exp ( - vN

2/ (2v,,2 )).

The temporal linear growth rates for Raman and Compton scattering can now be ob-

tained for the backscattered electromagnetic wave in the beam frame. The energy flux of the

incident pump will be assumed to propagate towards the right, i.e., w„ > 0 and k0 > 0, as

shown in Fig. (1).

Raman Scattering (Wave-Wave Scattering)

We first consider scattering off a cold electron distribution such that | co(, /k^ \ » vth

or £ » 1. In the case of a small amplitude pump field the electrostatic mode is very close

to being an eigenmode of the pump-free system. That is, for a small amplitude pump a>n and

Jt|'| approximately satisfy the dispersion relation 1 + x (u>^ , k^ ) = 0. If the pump amplitude

is large enough, the eigenmodes of the electrostatic wave are modified and no longer satisfy the

relationship given by 1 + \ '(tü|j , k\\ ) — 0. This strong pump regime will be discussed later.

The dispersion relation in Eq. (18) leads to unstable roots if D'_ or D+ vanish simultaneously

along with the left hand side of the equation. Figure (2) shows the general form of the disper-

sion relation in Eq. (18) for a cold electron system and small amplitude pump field. The situa-

tion where both £>'_ or D+ vanish simultaneously will not be considered here since this case

does not correspond to stimulated backscattering; and hence, will not lead to the proper fre-

11

SPRANGLE AND DROBOT

quency enhancement in the laboratory frame. Furthermore, the linear growth rate is substan-

tially smaller for this instability. Because the frequency of the longitudinal wave is much less

than the frequency of the pump wave, the quantity D± (OJ±,A± ) can be approximated. With

this assumption we find that

D± ((D±,k± ) = ± 2wj Mil c" cV

+ (20)

UNSTABLE ROOT

' D'+iu>\, k'

tu' , k'..l = 0

UNSTABLE ROOT

Figure 2. Dispersion Relation in the Beam Frame, for a Cold Electron Beam, Showing Stimulated Growth of the Scat- tered Radiation.

In obtaining (20) the fact that the pump wave satisfies the dispersion relation, D() (o>0,k0)

= 0, was also used. Since we are considering scattered waves such that |&i||/A|j | » vlh,

the susceptibility can be expanded to give x — ~^p2/^\\2 ~ 3üJ,i2vih k\\'lu,\\^ + Hm(x')

where Im(x') = n 1/2 i exp ( —£ 2) is the imaginary part of the electron susceptibility. From

here on we will take the wave at (w _, k _) to be resonant, i.e. D'_ (w '_, k'_) = 0, and the

(w +, k + ) wave to be nonresonant, i.e. D+ (w +, k + ) ^ 0. Therefore, we consider the case

where o)0/ko > 0 and w _/k _ < 0. Our choice for the resonant backscattered wave,

D _ = 0, is completely arbitrary, since it is easy to see that choosing D+ = 0 and D _ ^0

leads to the same results. The dispersion relation in (18) now becomes

"\ I-* ii i ULI n

(a;,,2 -W/2(l -/7m(x ))(w|| -«)=--£

o

12

2 v(;2A,i2

(21)

NRL MEMORANDUM REPORT 3587

where

n '= A:>H c2/(o0 - c2/cn2/(2o)0) and w/2 = a>p

2 + 3vth2^j2 = wp

2.

Equation (21) is the approximate form of the dispersion relation when 1 + x — 0, D_— 0,

D'+ T£ 0, £ ' » 1 and w. « w^. From Fig. (2) we note that the unstable roots occur for

/C|j = 2 Ar0, which corresponds to stimulated backscattering.

To obtain the growth rate from Eq. (21) we set wn = w, + 8o> where to, is set equal to

a'and 18w'| as well as |&J//m(x ')I are assumed much less than |w/|. Substituting

ct)|j = w/ + Söj in Eq. (21) gives the following expression for 8w

'2 '2; '2 ) ^2

(mjlm^x'))2 + w/ ' i

8w = — /— //w (x ) + y to ,&) lwo (22)

There are two cases to consider in Eq. (22), depending on the strength of the incident pump

wave. If the incident pump amplitude is sufficiently weak to satisfy the inequality

1/2

ßos « (»;3«oi/2

lm(x ) VNI

Im(x ) (23)

where ßos = vos/c, then the temporal growth rate is given by

Im (8co ) = — >pvosk\\

>/lm(x ) //w(x') _ £

/w(x ) (24)

and the real part of the frequency is /?e(w|j ) = w/ — oy In the moderately strong pump re-

gime where ßos » (w/3w0)1/2 /m(x ')/(apck\\ ), but small enough so as not to greatly

modify the pump-free eigenmodes, the growth rate is

1/2

w//m(x ) r' = lm(6(o') =v 4

V<«"ll

V^/^o £«

2 (25)

13

SPRANGLE AND DROBOT

the real part of the frequency is

'2

w '2 (26)

We now consider the situation where the pump amplitude is so large that it modifies the

eigenmodes of the electrostatic waves. That is, if ßos > /3c'rit where /3c'rit = (2 u',l<a'0) m -

(2(op/wg)1/2. The frequency and wavenumber of the longitudinal wave, (w|J , k\\ ), no longer

satisfies the relationship 1 + \ '(<»\\ . k\\) — 0. In this case W|j >> cu/,0 'and the disper-

sion relation in (21) takes the form wM3 = - {ßos<x)p)

2o)'0l2 which gives the growth rate

r'=/m(W|j) =^((0^)2^/2)1/3. (27)

The real part of the frequency in this case is

Re(o,u) =| ((/S^)2^)173. (28)

Equations (24), (25) and (27) are the expressions for the temporal growth rates for stimu-

lated Raman backscattering in the beam frame. These expressions all have a different

parametric dependence on the pump amplitude. Since lm{\ ) « 1, for a cold electron distri-

bution, we set lm(x ) =0 and discuss only the moderately strong and strong pump regimes

whose growth rates are given in Eqs. (25) and (27) respectively. The results of the linear

theory for these two cases can easily be transformed back to the laboratory frame. The value

of ßos, in the laboratory frame, which distinguishes the moderately strong and strong pump re-

gime is /3crit, and is given by

1/2

'cnt = ,, -3/2 r<T3/2Ü +ß0)-

l/2

y-"H\ +/?0)-1/2 -phln

<ti,

1/2

(29)

14

NRL MEMORANDUM REPORT 3587

where ß = v0/c, y0 = (1 - (v0/c)2) ~1/2 and v0 is the axial speed of the beam in the

laboratory frame. From Eq. (25) and (27) we find that the linear growth rate in the laboratory

frame, for the moderately strong and strong pump regime, is given respectively by

when ßos < ßcrit and

r -^ (d +ß0)ßl<o0»t/2)U^^f- (d +ß0)ß2o^0-2

Pho/2)m. (3i)

when ß > ß •,. The frequency of the backscattered electromagnetic wave in both the * os ^ r 11

|w_| = (1 + ß0)2y0

2«v

'os ^ ^cnf

above cases is \„> I = (1 + ßJ2y2<o„.

(32)

In the beam frame of reference the phase velocity of the electrostatic wave is

«o,'| /A:,', -ü>//2/c0) (33)

when ßos < ßc'rit and

«ii/*ii -| ((ßos«;)2^0/2)1/3//c0 > C0//2C

for ß ' > ß ' •,. The growth rates in Eqs. (30) and (31) are valid as long as | w,j //c,j | is much • os c-ni

greater than the thermal velocity vth. The opposite limit is the Compton regime and will be

discussed in detail later. The thermal velocity in the beam frame is related to the thermal velo-

city in the laboratory frame by the relation vth = y02vth. The total spread in the beam energy

in the laboratory frame due to the thermal velocity spread vth is A€th = 2ß0y0 (vth/c)mc.

Therefore, for thermal effects to be negligible and Eq. (30) and (31) applicable, the following

conditions on Aeth must be satisfied in the moderately strong pump case

«Phl12 A / ß° Aeth/e° << (1 +ß0)(y0 -1) (35)

15

SPRANGLE AND DROBOT

and in the strong pump case

Aeth/e0 « y0ßtA

(1 +ß0)(y0 -1) ßi w/./v<»

1/2 (1 +ßo)

1/3

(36)

where e(1 = (y(, — l)m0c2 is the electron kinetic energy. These results are summarized in

Table I for a highly relativistic electron beam. Estimates for the efficiency of converting elec-

tron beam energy into electromagnetic energy are also given in Table I and will be discussed

shortly.

Compton Scattering (Wave-Particle Scattering)

We now consider the kinetic regime where the phase velocity of the longitudinal wave is

of the order of the electron thermal velocity, i.e., wi| /A,, =#(vlh ). In this regime the elec-

trostatic waves are heavily Landau damped in the absence of the pump wave. This scattering

mode is called stimulated Compton or inverse nonlinear Landau scattering because the elec-

trostatic wave, resulting from the beating of the two electromagnetic waves, is resonant with

the electrons. Since the longitudinal wave is not an eigenmode of the system, i.e., 1 + \ 5^0,

the dispersion relation of the electrostatic wave for Compton backscattering takes the form

- n = (j3ft2/2)w,;*/(! + v), (37)

where Eq. (20) for D_ together with k^ — 2k0 = 2co0/c were used in obtaining Eq. (37).

Taking the imaginary part of both sides of Eq. (37) and noting that

Im 1 + X

- Im 1

1 + X

the growth rate for Compton scattering, in the beam frame, is found to be

r = - (ß02/2)ojnlm

1

1 + X '(n \2k0) (38)

16

NRL MEMORANDUM REPORT 3587

where

/»/[(l + x (n ,2k0)) ~[] = -h =.

lm(Ze)

'-T

2

e{Zc) + j_ *Z> 2 k

Im(Zc)

Ze = dZ/B£ , and £ = w / (\/2vlhA: ). The term /w((l + \ ) _1) can be readily approxi-

mated in the limit that <u Ik « vlh,i.e., £ « 1. In this domain the wavelength of the

electrostatic disturbance, |2<r/A- |, is much less than the shielding length, 2-/kD. Therefore,

in the limit that kD/k << 1 and

2

OR. /W(7.)== -,i:^ 2

/,„ [(1 +x ) -']| - j An A << 1

£ exp( -£ 2). (39)

Substituting Eq. (39) into (38). the Compton growth rate becomes

1 2

-A.:«"« £ exp( -£ -).

The term £ exp ( —£ -) has a maximum when £ = \ \2. i.e.. & ,k

imum value of T in Eq. (40) is approximately given by

rmax = A '"A- {kD'k )2 "JQ

(40)

= vlh, so the max-

' ih (41)

In ref. (8) it was shown that the temporal Compton growth rate has the following transforma-

tion properties from the beam frame to the laboratory frame.

r r (1 + v„. c) (42)

17

SPRANGLE AND DROBOT

Substituting Eq. (41) into (42) and writing the beam frame quantities in terms of laboratory

frame quantities we obtain

r --i ^ max 10 -.5 y0s(l +ß0)

2cn 'th

2 ßho^l 5 o +ß0)

0« eo

2-f) (y„ -i)2 Aeth

l ^2 0,» eo 2

10 w0 r„ Aelh

(43)

where the last expression is valid for a highly relativistic electron beam.

Section III. Saturation Levels and Efficiencies

This section will deal primarily with the saturation and efficiency levels of Raman back-

scattering off a cold, i.e., vth =0, electron beam. Saturation of the backscattered electromagnetic

wave may be due to either pump depletion or nonlinearities associated with the electrostatic

wave (density wave). Pump depletion ocurrs when the amplitude of the pump is depleted by

the scattering process. Nonlinearities result when the electrostatic wave, given in Eq. (2),

grows to a level sufficient to trap electrons. Roughly speaking, for a small amplitude pump

field, pump depletion occurs before the electron dynamics become nonlinear. However, for a

large amplitude pump field electron trapping takes place before all the incident photons are

scattered. Therefore, the magnitude of ßos determines the nature of the saturation mechanism.

In the beam frame the magnitude of the backscattered electromagnetic wave, when sa-

turation is due to pump depletion, is given by

E'-\ = (a/_/o'%,; o' ^o-

(44)

NRL MEMORANDUM REPORT 3587

Equation (44) is just a statement of conservation of wave action. When the frequency of the

scattered wave, u> _, is approximately equal to the pump frequency, w(), in the beam frame, we

find that |£_| = £„ and virtually all the incident pump photons are backscattered. However,

before this happens the level of the density wave may become comparable to the ambient den-

sity of the electrons. When this happens the electron dynamics become nonlinear and electron

trapping occurs in the potential well associated with the total electrostatic field. The total longi-

tudinal electrostatic field consists of the sum of the self consistent field given by Eq. (15a) and

the ponderomotive field associated with the v x B/c axial force. The magnitude of the sum of

these two fields is

Emnil\ ~ 1 A"ll X

2" (1 +X)

\e\ E E

(45)

Associated with | E. . ,\ is a density wave, the magnitude of which is

8/; *ll X E„E_

87r (1 + X ) '"„<%&> (46)

Equating |8/? | to the ambient electron density n0, we find that electron trapping limits |£_|

to the value

£- = 8T«„'»„ (1 + x ') w»w

k,/E.. (47)

For the moderately strong pump regime we find that

1 +X

X |2I'/o

ßos

1/2

(48)

where Eq. (25) was used for I' . In the case of a strong pump the magnitude of the susceptibil-

ity is much less than unity and therefore

19

SPRANGLE AND DROBOT

i + x' = 1

X =

W|| 2

=: ßos "o

X «V 2u>n

2/3

(49)

where Eqs (27) and (28) were used for wN . Substituting these expressions for \ '/0 + X )

into Eq. (47) we find that the amplitude of the backscattered electromagnetic wave, when sa-

turation is due to electron trapping, for the moderate and strong pump case is respectively

given by

3/2 E<> ßos < Pcri. ß, (50a)

and

1 2 5/3

<»o

4/3

(ßos) 2/3 ' ßos > ßcril

(50b)

where we have used the fact that A,, = 2ka = 2a>0/c and w '_ = oV Comparing Eq. (50a)

with (44) we find that if ßm < 0,', where j8,' = (1/2) (w/wj 3/2, then pump depletion sa-

turates the backscattering process before electron trapping takes place. Since 0, is always less

than ßcril it is clear that for ßm > 0, electron trapping is the saturation mechanism and it oc-

curs before the pump is depleted. The level of the fields at saturation in the beam frame can

then be summarized as follows

\E_\ =

1

for ßm < ß], pump depletion

for0l' < ßi,s < j8cri,. trapping

-jf <ß\/ßer\i) (ßcr-Jß,J2,'3E,,- for jÖm. > j8cril. trapping. (51a-c)

In order to obtain the efficiency it is necessary to transform the magnitudes of the backscat-

tered fields in Eqs. (51) to the laboratory frame. Since the electric fields have the following

transformation properties, | £_ | =|£_|/(r„(l +0„))and£„ = (1 + ß0 )y<)E0, the ampli-

tudes in Eqs. (51) when written in the laboratory frame become

20

NRL MEMORANDUM REPORT 3587

,E-\ = (1 +/3())27o

E„.

Pi

ßas £„•

1 01 72 0cril

Jcnl 2/3

for/8m. < /3,

for/8, < /BttV < 0cril

for0m. > 0cril

(52a-c)

where

01 =p»"5/2(1 +/3"} -3/2 '>,,hn

1/2 3/2

Peri, =21/2r,r3/2(l +ßo) -1/2 y/y»

1/2

and

1/2

are the expressions for jo,' and /3cril transformed to the laboratory frame.

The efficiency of stimulated backscattering can be defined as the ratio of the average elec-

tromagnetic energy density in the laboratory frame to the kinetic energy density of the elec-

trons. Efficiency is then defined as T, = < WE + WM >/(n„(y„ - l)mt,c2) where

< WE> — < WM> = |£2_|/167r is the average electric field energy density. The electric

energy density is very nearly equal to the magnetic energy density, since ck _ = u _. Using

the expressions in Eqs. (52) for |£_|, the efficiencies in the three regimes determined by the

magnitude of ßos are given by

2

1 y050 + 0„>4

2 (y„ -1) 'pi > o

1/2 ßos-

1 (1 + 0O) o>Jy}/2

16 (y„ -1)

up/y„ ßl

«2

1/3

forj3m. < ßx

for/3, < 0<M < ßcra

forj8,v > /3cril. (53a-c)

21

SPRANGLE AND DROBOT

Table I contains a summary of the results obtained for the Raman backscattering instability off

a highly relativistic electron beam. It contains the pump amplitude regimes in terms of ß

temporal growth rates, saturation mechanisms, efficiencies at saturation and energy spread re-

quirements.

Table I. Summary of Collective Wave-Wave Scattering Results in the Laboratory Frame for a Highly Relativistic Electron Beam. The parameters are defined as

ß0 = l.ßm =\e\E0/(yomi)c0c),ßl =(32r1/V5'V/2,/3crit =y-"Hm, £ = (o>„h !/2) Zealand \e Je () = 2y3„ (wjc)/(y0 -1).

Regime Growth Rate Saturation Mechanism

Saturation Efficiency

Energy Spread Requirements

0<ßos<ß! r = _^7i/2 a/2 V2 ° °

pump depletion

8?0 „ (17 /f\2 Aeth t

« ?

e0 2(7o - 1) 1 „ -1 (Pos/S) • o x

ßl<ßos<ßcnt ßoi

trapping n- S Aeth t « 5

e0 2(7o -1) " 4(To - 1)

ßent <ßos r=^«„(ftji)2'3 trapping "4J°-i««»2's A^th ^ T0(ßosf)

2/3

e0 ^ 4(7o-l)

22

NRL MEMORANDUM REPORT 3687

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23